If you have a CHA voucher, you will need to do a portability to HACC, the cook county authority or another authority in the suburbs. Porting out takes a couple of months usually. In practice, it takes 2-3 weeks for CHA to process your paperwork and then give to new authority. Then it takes them a little while to process you and get you a briefing date to get your moving papers. Usually another 2-3 weeks. Keep that timetimable in mind. You should not start looking for a place until the new authority at least has scheduled your briefing. Otherwise, there is too much of a lag time as it takes several weeks once a landlord has signed moving papers to have an inspection and move.
You can only port 1x per year, so you need to choose the right county or authority to move into. And keep in mind that you will not have the same voucher amount always when you port. Different authority sometimes different rules. I do have a leasing agent who can help you once you get your briefing date.